Finite Group Databasebeta

← #14326839

Subgroups — #14326839

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91781C211yesmaximal, characteristic
2C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
4#11748867C212yesmaximal, complemented
5#11748858C211yesmaximal, characteristic, complemented
6#11748854C212yesmaximal
7#11748841C211yesmaximal, characteristic, complemented
8#11748829C212yesmaximal
9#11748799C211yesmaximal, characteristic
10C4×#6667C211yesmaximal, characteristic
11#11748611C212yesmaximal, complemented
12#10605364—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×#6714C2212yes—
16C2×C4⋊Dic15C2211yescharacteristic
17C2×C4×Dic15C2212yes—
18#91781C418yescomplemented
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2212yes—
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4×Dic5C2212yes—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×#6669C2212yes—
26C2×#6669C2212yes—
27C2×#6667C2211yescharacteristic
28C2×#6667C2211yescharacteristic
29C2×#6667C2212yes—
30C5×C3⋊#95C2212yes—
31C3×#3963C2212yes—
32#90994C2211yescharacteristic
33#90994C2212yes—
34#90994C2212yes—
35#90994C2212yes—
36#90994C2211yescharacteristic
37#73633—51nomaximal, Hall
38C2×#4038—31no—
39C4×C5×C4⋊C4—31no—
40#65276—31no—
41C4⋊C4×Dic5D311yescharacteristic, complemented
42#65262—31no—
43#65229—32no—
44C4×#685—31no—
45#65221—32no—
46#65217—31no—
47#64958—32no—
48#64939—32no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2311yescharacteristic
53C22×Dic15C2312yes—
54C2×C4×D15—22no—
55C2×C4×D15—22no—
56C2×C4×D15C2×C414yes—
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C22×C3×Dic5C2311yescharacteristic
63C3×C5×C4⋊C4C2×C414yes—
64#6714C2×C418yes—
65C4⋊Dic15C2×C414yes—
66C4×Dic15C2×C418yes—
67C5×C4⋊Dic3—22no—
68C4×C5×Dic3—22no—
69C4×C5×Dic3—24no—
70C3×C4⋊Dic5—22no—
71C3×C4×Dic5—22no—
72C3×C4×Dic5—24no—
73#6669—42no—
74#6669—42no—
75#6667—22no—
76#6667—22no—
77#6667—42no—
78C2×#892—51no—
79C3×C4×C4⋊C4—51no—
80#4987—51no—
81Dic3×C4⋊C4D511yescharacteristic, complemented
82#4972—51no—
83#4940—52no—
84C4×(C2×D3)⋊C4—51no—
85#4932—52no—
86#4928—51no—
87#4669—52no—
88#4650—52no—
89C22×C4×D5—31no—
90C2×C5×C4⋊C4—31no—
91C2×C5×C4⋊C4C2×D311yescharacteristic
92C2×C42×C5—31no—
93C2×C42×C5—32no—
94C2×#685—32no—
95C2×#685—31no—
96C2×#685—31no—
97C2×#685—32no—
98C2×C4⋊Dic5—31no—
99C2×C4⋊Dic5C2×D311yescharacteristic
100C2×#683—32no—
101C2×#683—32no—
102C2×C4×Dic5—32no—
103C2×C4×Dic5C2×D312yes—
104C2×C4×Dic5C2×D311yescharacteristic
105#4038—38no—
106C5×#95—32no—
107#3963—31no—
108#3963—32no—
109#3963—32no—
110#3963—32no—
111#3963—31no—
112#3963C2×D312yes—
113#2346—151noSylow
114C23×C3×C5C2411yescharacteristic
115C22×D15—21no—
116C22×D15—21no—
117C22×D15—21no—
118C22×D15—21no—
119C22×D15—21no—
120C22×D15—21no—
121C22×D15C22×C412yes—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5C22×C414yes—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5#4112yes—
127C2×C3×C4×C5#4112yes—
128C2×Dic15—21no—
129C2×Dic15—21no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15C22×C412yes—
133C2×Dic15C22×C414yes—
134C4×D15—28no—
135C2×C5×Dic3—21no—
136C2×C5×Dic3—21no—
137C2×C5×Dic3—21no—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3#4112yes—
144C2×C3×Dic5—21no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5#4112yes—
153C22×C4×D3—51no—
154C2×C3×C4⋊C4—51no—
155C2×C3×C4⋊C4C2×D511yescharacteristic
156C2×C3×C42—51no—
157C2×C3×C42—52no—
158C2×(C2×D3)⋊C4—52no—
159C2×(C2×D3)⋊C4—51no—
160C2×(C2×D3)⋊C4—51no—
161C2×(C2×D3)⋊C4—52no—
162C2×C4⋊Dic3—51no—
163C2×C4⋊Dic3C2×D511yescharacteristic
164C2×Dic3⋊C4—52no—
165C2×Dic3⋊C4—52no—
166C2×C4×Dic3—52no—
167C2×C4×Dic3C2×D512yes—
168C2×C4×Dic3C2×D511yescharacteristic
169#892—58no—
170C3×#95—52no—
171C3⋊#95—51no—
172C3⋊#95—52no—
173C3⋊#95—52no—
174C3⋊#95—52no—
175C3⋊#95—51no—
176C3⋊#95C2×D512yes—
177C23×D5—31no—
178C22×C4×C5—31no—
179C22×C4×C5—32no—
180C22×C4×C5—31no—
181C22×C4×C5C22×D312yes—
182C22×C4×C5C22×D311yescharacteristic
183C22×Dic5—31no—
184C22×Dic5—32no—
185C22×Dic5C22×D312yes—
186C22×Dic5C22×D311yescharacteristic
187C22×Dic5C22×D311yescharacteristic
188C2×C4×D5—34no—
189C2×C4×D5—62no—
190C2×C4×D5—62no—
191C5×C4⋊C4—62no—
192C5×C4⋊C4C4×D314yes—
193C42×C5—64no—
194C42×C5—62no—
195#685—38no—
196#685—62no—
197#685—62no—
198#685—122no—
199C4⋊Dic5—22no—
200C4⋊Dic5—34no—
201#683—122no—
202#683—122no—
203C4×Dic5—22no—
204C4×Dic5—24no—
205C4×Dic5—38no—
206C2×#117—151no—
207#389—151no—
208#389—151no—
209#388—152no—
210#388—152no—
211#382—152no—
212#380—151no—
213C4×C4⋊C4—31no—
214C4×C4⋊C4—51no—
215C4×C22⋊C4—151no—
216#376—152no—
217C22×C3×C5C23×C411yescharacteristic
218C22×C3×C5C2×#4111yescharacteristic
219C22×C3×C5C2×#4111yescharacteristic
220C22×C3×C5C2×#4111yescharacteristic
221C22×C3×C5C2×#4111yescharacteristic
222C22×C3×C5H(5,𝔽2)11yescharacteristic
223C22×C3×C52−1+411yescharacteristic
224C2×D15—22no—
225C2×D15—22no—
226C2×D15—22no—
227C2×D15—41no—
228C2×D15—41no—
229C2×D15—41no—
230C2×D15—41no—
231C3×C4×C5—24no—
232C3×C4×C5#11714yes—
233Dic15—22no—
234Dic15—24no—
235C3×Dic5—22no—
236C3×Dic5—42no—
237C3×Dic5—41no—
238C5×Dic3—22no—
239C5×Dic3—41no—
240C5×Dic3—42no—
241C23×D3—51no—
242C22×C3×C4—51no—
243C22×C3×C4—52no—
244C22×C3×C4—51no—
245C22×C3×C4C22×D512yes—
246C22×C3×C4C22×D511yescharacteristic
247C22×Dic3—51no—
248C22×Dic3—52no—
249C22×Dic3C22×D512yes—
250C22×Dic3C22×D511yescharacteristic
251C22×Dic3C22×D511yescharacteristic
252C2×C4×D3—54no—
253C2×C4×D3—102no—
254C2×C4×D3—102no—
255C3×C4⋊C4—102no—
256C3×C4⋊C4C4×D514yes—
257C3×C42—104no—
258C3×C42—102no—
259(C2×D3)⋊C4—58no—
260(C2×D3)⋊C4—102no—
261(C2×D3)⋊C4—102no—
262(C2×D3)⋊C4—202no—
263C4⋊Dic3—22no—
264C4⋊Dic3—54no—
265Dic3⋊C4—202no—
266Dic3⋊C4—202no—
267C4×Dic3—22no—
268C4×Dic3—24no—
269C4×Dic3—58no—
270C23×C5C23×D311yescharacteristic
271C22×D5—32no—
272C22×D5—61no—
273C22×D5—61no—
274C22×D5—61no—
275C22×D5—61no—
276C22×D5—61no—
277C22×D5—61no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—32no—
281C2×C4×C5—61no—
282C2×C4×C5—61no—
283C2×C4×C5—61no—
284C2×C4×C5—62no—
285C2×C4×C5—61no—
286C2×C4×C5—62no—
287C2×C4×C5—61no—
288C2×C4×C5—62no—
289C2×C4×C5C2×C4×D314yes—
290C2×C4×C5C2×C4×D312yes—
291C2×C4×C5#23512yes—
292C2×C4×C5#23512yes—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—21no—
299C2×Dic5—21no—
300C2×Dic5—22no—
301C2×Dic5—32no—
302C2×Dic5—34no—
303C2×Dic5—61no—
304C2×Dic5—61no—
305C2×Dic5—62no—
306C2×Dic5—62no—
307C2×Dic5#23712yes—
308C4×D5—68no—
309C23×C4—151no—
310#117—158no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—51no—
313C2×C4⋊C4—151no—
314C2×C4⋊C4—152no—
315C2×C4⋊C4—152no—
316C2×C4⋊C4D3×D511yescharacteristic
317C2×C22⋊C4—152no—
318C2×C22⋊C4—151no—
319C2×C22⋊C4—151no—
320C2×C22⋊C4—152no—
321C2×C42—32no—
322C2×C42—31no—
323C2×C42—52no—
324C2×C42—51no—
325C2×C42—152no—
326#95—32no—
327#95—52no—
328#95—151no—
329#95—152no—
330#95—152no—
331#95—152no—
332#95—151no—
333C2×C3×C5C2×#11711yescharacteristic
334C2×C3×C5C4×#4111yescharacteristic
335C2×C3×C5#52011yescharacteristic
336C2×C3×C5#55211yescharacteristic
337C2×C3×C5#55211yescharacteristic
338C2×C3×C5#55511yescharacteristic
339C2×C3×C5#55511yescharacteristic
340D15—42no—
341C23×C3C23×D511yescharacteristic
342C22×D3—52no—
343C22×D3—101no—
344C22×D3—101no—
345C22×D3—101no—
346C22×D3—101no—
347C22×D3—101no—
348C22×D3—101no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—52no—
352C2×C3×C4—101no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—102no—
356C2×C3×C4—101no—
357C2×C3×C4—102no—
358C2×C3×C4—101no—
359C2×C3×C4—102no—
360C2×C3×C4C2×C4×D514yes—
361C2×C3×C4C2×C4×D512yes—
362C2×C3×C4#70912yes—
363C2×C3×C4#70912yes—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—52no—
373C2×Dic3—54no—
374C2×Dic3—101no—
375C2×Dic3—101no—
376C2×Dic3—102no—
377C2×Dic3—102no—
378C2×Dic3#71112yes—
379C4×D3—108no—
380C22×C5C22×C4×D311yescharacteristic
381C22×C5C2×#23511yescharacteristic
382C22×C5C2×#23511yescharacteristic
383C22×C5C2×#23711yescharacteristic
384C22×C5#100711yescharacteristic
385C22×C5D3×#4111yescharacteristic
386C22×C5#100911yescharacteristic
387C2×D5—62no—
388C2×D5—62no—
389C2×D5—62no—
390C2×D5—121no—
391C2×D5—121no—
392C2×D5—121no—
393C2×D5—121no—
394C4×C5—24no—
395C4×C5—62no—
396C4×C5—122no—
397C4×C5—121no—
398C4×C5#87214yes—
399Dic5—22no—
400Dic5—41no—
401Dic5—42no—
402Dic5—62no—
403Dic5—64no—
404C24—151no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—32no—
408C22×C4—51no—
409C22×C4—51no—
410C22×C4—52no—
411C22×C4—151no—
412C22×C4—152no—
413C22×C4—154no—
414C22×C4—302no—
415C22×C4—302no—
416C22×C4C2×D3×D512yes—
417C22×C4C2×D3×D511yescharacteristic
418C4⋊C4—62no—
419C4⋊C4—102no—
420C4⋊C4—154no—
421C4⋊C4—602no—
422C4⋊C4—602no—
423C4⋊C4#120014yes—
424C22⋊C4—158no—
425C22⋊C4—302no—
426C22⋊C4—302no—
427C22⋊C4—602no—
428C42—64no—
429C42—62no—
430C42—104no—
431C42—102no—
432C42—158no—
433C3×C5#234611yescharacteristic, complemented, Hall
434C22×C3C22×C4×D511yescharacteristic
435C22×C3C2×#70911yescharacteristic
436C22×C3C2×#70911yescharacteristic
437C22×C3C2×#71111yescharacteristic
438C22×C3#414711yescharacteristic
439C22×C3D5×#4111yescharacteristic
440C22×C3#414911yescharacteristic
441C2×D3—102no—
442C2×D3—102no—
443C2×D3—102no—
444C2×D3—201no—
445C2×D3—201no—
446C2×D3—201no—
447C2×D3—201no—
448C3×C4—24no—
449C3×C4—102no—
450C3×C4—202no—
451C3×C4—201no—
452C3×C4#401814yes—
453Dic3—22no—
454Dic3—41no—
455Dic3—42no—
456Dic3—102no—
457Dic3—104no—
458C2×C5C2×#87211yescharacteristic
459C2×C5#551311yescharacteristic
460C2×C5#553211yescharacteristic
461C2×C5C4×#23711yescharacteristic
462C2×C5#556311yescharacteristic
463C2×C5#557411yescharacteristic
464C2×C5#558111yescharacteristic
465D5—122no—
466C23—152no—
467C23—301no—
468C23—301no—
469C23—301no—
470C23—301no—
471C23—301no—
472C23—301no—
473C23C22×D3×D511yescharacteristic
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—32no—
477C2×C4—52no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—62no—
482C2×C4—61no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—61no—
486C2×C4—101no—
487C2×C4—101no—
488C2×C4—101no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4—101no—
492C2×C4—102no—
493C2×C4—101no—
494C2×C4—152no—
495C2×C4—154no—
496C2×C4—301no—
497C2×C4—301no—
498C2×C4—302no—
499C2×C4—302no—
500C2×C4—308no—
501C2×C4#677112yes—
502C2×C4#677112yes—
503C2×C4C2×#120014yes—
504C2×C4C2×#120012yes—
505C2×C3C2×#401811yescharacteristic
506C2×C3#6584711yescharacteristic
507C2×C3#6586611yescharacteristic
508C2×C3C4×#71111yescharacteristic
509C2×C3#6589711yescharacteristic
510C2×C3#6590811yescharacteristic
511C2×C3#6591511yescharacteristic
512D3—202no—
513C5#7363311yescharacteristic, complemented, Sylow
514C22—302no—
515C22—302no—
516C22—302no—
517C22—601no—
518C22—601no—
519C22—601no—
520C22—601no—
521C22C2×#677111yescharacteristic
522C22C2×#677111yescharacteristic
523C22D5×#23711yescharacteristic
524C22D3×#71111yescharacteristic
525C22#9202911yescharacteristic
526C22#9203511yescharacteristic
527C22C22×#120011yescharacteristic
528C4—24no—
529C4—62no—
530C4—102no—
531C4—121no—
532C4—122no—
533C4—202no—
534C4—201no—
535C4—302no—
536C4—304no—
537C4#9138614yes—
538C3#1060536411yescharacteristic, complemented, Sylow
539C2—602no—
540C2#1175334711yescharacteristic
541C2#1175335611yescharacteristic
542C2#1175373611yescharacteristic
543C2#1175376711yescharacteristic
544C2C2×#9138611yescharacteristic
545C2#1175485311yescharacteristic
546C2#1175514911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2346
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C5×Dic3×C4⋊C4C2semidirect
3. C3×C4⋊C4×Dic5C2semidirect
4. #11748867C2semidirect
5. #11748858C2semidirect
7. #11748841C2semidirect
11. #11748611C2semidirect
18. #91781C4semidirect
41. C4⋊C4×Dic5D3semidirect
81. Dic3×C4⋊C4D5semidirect
433. C3×C5#2346semidirect
513. C5#73633semidirect
538. C3#10605364semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#917812
C5×Dic3×C4⋊C42
C3×C4⋊C4×Dic52
#117488672
#117488582
#117488542
#117488412
#117488292
#117487992
C4×#66672
#117486112
#106053643
#736335